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Revista brasileira de fisica

A new semiclassical approach for the spatial density of nuclear systems

D. Galetti, S Pawell

Published Jan 1, 1990• 0 citations
OpenAlex

Abstract

We show that a new decomposition of the many-body Wigner function in the squeezed states basis set of the quantum phase-space as- sociated to bound states allows one to separate the spatial density of the many-body system into a semiclassical part plus its quantum complement.

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On the Center-of-Mass Free Density Matrix

Kazuhiro Yabana

Two methods to calculate the density matrix of one• body or more•than•one•body which does not include the spurious center•of•mass excitation are presented.One is the method to calculate the density matrix for the many-body wave function of general type and the other is applicable to the case in which the center~of-mass motion separates from the internal motion and is in the (Os) orbit of harmonic oscillator.The applications of these methods are illustrated for some many-body wave function.§ 1. IntroductionTo represent the characteristic features of the many-body wave function of nuclei, the density matrices are often utilized.Experimentally, for example, in the electron scattering from nuclei, the nuclear structure is analyzed by the terms of the nuclear density and the current density, which are the special forms of the one-body density matrix.Theoretically, the one-body density matrix plays a crucial role in the current nuclear theories based on the mean field approximation, such as Hartree-Fock theory.And, in recent nuclear studies, interest is taken in not only the nuclear density which is the diagonal part of the one-body density matrix, but the non-local part of it.The non-local part of the one-body density matrix is often visualized with the use of the Wigner transform of it. 1),2)Especially, in the time-dependent Hartree-Fock theory, the Wigner transform of the one-body density matrix is interpreted as the classical distribution function and obeys the Vlasov equation approximately.3)These works clarify the importance of the non-local part of the one-body density matrix in the dynamics.Usually the spatial coordinate of the density matrix is measured from some spatially fixed origin.Due to the finiteness of the nuclear system, however, we need to measure the coordinate from the center of mass of the system; otherwise the spurious center-of-mass motion affects the density matrix.This effect of center-of-mass motion becomes important for light systems.For the diagonal part of the one-body transition density matrix, the effect of center-of-mass motion was discussed by Tassie and Barker for the case of harmonic oscillator shell mode1. 4) And for the diagonal part of the one-body density matrix in momentum representation, that is the nuclear momentum distribution, the effect of spurious center-of-mass motion has been discussed. 5) For the general non-local density matrix, however, there seems not to be presented any method to treat the spurious center-of-mass effect.In this paper, we present two methods to get the center-of-mass free density matrix.One is the method, applicable to the general wave function, which is especially suitable for the case where the wave function is a (linear superposition of) Slater determinant.The other is applicable to the cases in which the center-of-mass motion separates from other internal motions and is in the (Os) orbit of harmonic oscillator.For such wave functions, the latter method is more easy to calculate though the former method is also applicable.

Progress of Theoretical Physics•Aug 1, 1986•4 citations
In this work we study the Wigner functions, which are the quantum analogues of the classical phase space density, and show how a full rigorous semiclassical scheme for all orders of ℏ can be constructed for them without referring to the actual coordinate space wavefunctions from which the Wigner functions are typically calculated. We find such a picture by a careful analysis around the stationary points of the main quantization equation, and apply this approach to the harmonic oscillator solving it for all orders of ℏ.
In this paper, we study within the structure of Symplectic Quantum Mechanics a bidimensional nonrelativistic strong interaction system which represent the bound state of heavy quark-antiquark, where we consider a Cornell potential which consists of Coulomb-type plus linear potentials. First, we solve the Schrödinger equation in the phase space with the linear potential. The solution (ground state) is obtained and analyzed by means of the Wigner function related to Airy function for the c c ¯ meson. In the second case, to treat the Schrödinger-like equation in the phase space, a procedure based on the Bohlin transformation is presented and applied to the Cornell potential. In this case, the system is separated into two parts, one analogous to the oscillator and the other we treat using perturbation method. Then, we quantized the Hamiltonian with the aid of stars operators in the phase space representation so that we can determine through the algebraic method the eigenfunctions of the undisturbed Hamiltonian (oscillator solution), and the other part of the Hamiltonian was the perturbation method. The eigenfunctions found (undisturbed plus disturbed) are associated with the Wigner function via Weyl product using the representation theory of Galilei group in the phase space. The Wigner function is analyzed, and the nonclassicality of ground state and first excited state is studied by the nonclassicality indicator or negativity parameter of the Wigner function for this system. In some aspects, we observe that the Wigner function offers an easier way to visualize the nonclassic nature of meson system than the wavefunction does phase space.
In this paper, we study within the structure of Symplectic Quantum Mechanics a bi-dimensional non-relativistic strong interaction system which represent the bound state of heavy quark-antiquark, where we consider a Cornell potential which consists of Coulomb-type plus linear potentials. First, we solve the Schrödinger equation in the phase space with the linear potential. The solution (ground state) is obtained and analyzed by means of the Wigner function related to Airy function for the cc‾c\overline{c}cc meson. In the second case, to treat the Schrödinger-like equation in the phase space, a procedure based on the Bohlin transformation is presented and applied to the Cornell potential. In this case, the system is separated into two parts, one analogous to the oscillator and the other we treat using perturbation method. Then, we quantized the Hamiltonian with the aid of stars operators in the phase space representation so that we can determine through the algebraic method the eigenfunctions of the undisturbed Hamiltonian (oscillator solution), and the other part of the Hamiltonian was the perturbation method. The eigenfunctions found (undisturbed plus disturbed) are associated with the Wigner function via Weyl product using the representation theory of Galilei group in the phase space. The Wigner function is analyzed, the non-classicality of ground state and first excited state is studied by the non-classicality indicator or negativity parameter of the Wigner function for this system. In some aspects, we observe that the Wigner function offers an easier way to visualize the non-classic nature of meson system than the wavefunction does.
Two methods to calculate the density matrix of one• body or more•than•one•body which does not include the spurious center•of•mass excitation are presented.One is the method to calculate the density matrix for the many-body wave function of general type and the other is applicable to the case in which the center~of-mass motion separates from the internal motion and is in the (Os) orbit of harmonic oscillator.The applications of these methods are illustrated for some many-body wave function.§ 1. IntroductionTo represent the characteristic features of the many-body wave function of nuclei, the density matrices are often utilized.Experimentally, for example, in the electron scattering from nuclei, the nuclear structure is analyzed by the terms of the nuclear density and the current density, which are the special forms of the one-body density matrix.Theoretically, the one-body density matrix plays a crucial role in the current nuclear theories based on the mean field approximation, such as Hartree-Fock theory.And, in recent nuclear studies, interest is taken in not only the nuclear density which is the diagonal part of the one-body density matrix, but the non-local part of it.The non-local part of the one-body density matrix is often visualized with the use of the Wigner transform of it. 1),2)Especially, in the time-dependent Hartree-Fock theory, the Wigner transform of the one-body density matrix is interpreted as the classical distribution function and obeys the Vlasov equation approximately.3)These works clarify the importance of the non-local part of the one-body density matrix in the dynamics.Usually the spatial coordinate of the density matrix is measured from some spatially fixed origin.Due to the finiteness of the nuclear system, however, we need to measure the coordinate from the center of mass of the system; otherwise the spurious center-of-mass motion affects the density matrix.This effect of center-of-mass motion becomes important for light systems.For the diagonal part of the one-body transition density matrix, the effect of center-of-mass motion was discussed by Tassie and Barker for the case of harmonic oscillator shell mode1. 4) And for the diagonal part of the one-body density matrix in momentum representation, that is the nuclear momentum distribution, the effect of spurious center-of-mass motion has been discussed. 5) For the general non-local density matrix, however, there seems not to be presented any method to treat the spurious center-of-mass effect.In this paper, we present two methods to get the center-of-mass free density matrix.One is the method, applicable to the general wave function, which is especially suitable for the case where the wave function is a (linear superposition of) Slater determinant.The other is applicable to the cases in which the center-of-mass motion separates from other internal motions and is in the (Os) orbit of harmonic oscillator.For such wave functions, the latter method is more easy to calculate though the former method is also applicable.